For the reign of Justinian we actually have some figures from the administrators Procopius and John Lydus. They have to be taken with a pinch of salt just because Romans were terrible economists, but Procopius in particular was excellently placed. He was Belisarius' secretary, which means he would have seen the accounts himself. I'm going to give figures in solidi, the main imperial currency, because that helps put things in perspective. 1lbs of gold was about 70 solidi and the annual imperial income in Justinian's reign varied between about 4.5-7m solidi.

Cost of war

The 1 year campaign in Africa (533–534) cost 7,000,000 solidi. That was an expedition of about 25,000 soldiers and supporting staff. Additional expenses came from having to get a fleet together to transport the soldiers, their horses, their weapons and food etc. To put that into perspective, the annual imperial income at the time was about 5-6,000,000 solidi. That campaign went really well and was relatively cheap, the following campaign in Italy did not go so well. The 13 year war between the Eastern Roman Empire and the Ostrogoths under Totila cost 1,600,000 solidi per year and a grand total of 300,000lbs of gold according to Procopius. The vast expense of this war is one of the main reasons Procopius utterly despised Justinian - he thought this was an enormous waste of time, men and money that should have been spent protecting the existing frontiers of the empire rather than creating new ones.

Cost of peace

Buying peace was far, far cheaper, not to mention the lives saved. Most Roman diplomatic payments went to Persia. At their peak, the payments ran over 11,000 pounds of gold or 770,000 solidi a year. At that time, that was about 13% of the empire's annual income. This was no small expense, but as you can see from the figures above it was about half as costly as a major war. Tiberius II (574–582) paid 7,200 pounds of gold per year for four years to get peace with Persia. These payments did not always work; a general rule about Persian treaties is that the Persians broke them when the oppurtunity was just too good, but they were still cheaper than a full-blown war between two of the world's most powerful states. As long as the money flowing to Perisa was more than the money they could get through looting Antioch, they wouldn't invade.

Other peoples were easier and cheaper to buy off. Justin II paid just 80,000 silver coins to the Avars for peace. Emperor Maurice paid them 80,000 gold solidi a year, which was still very little compared to the overall size of the economy. Tiberius II paid the Lombard tribes 200,000 solidi over several years to fight against each other to stop them unifying.

Overall, paying off the Persians was about half as costly as a war, and paying off barbarians was many times cheaper. This is in purely economic terms, not including the lives and infrastructure saved, which influenced this policy as well.